Comment on ‘Acoustic axes in weak triclinic anisotropy’ by Václav Vavryčuk

The existence of acoustic axes (singularities, degeneracies) is a common feature in anisotropic media. Along acoustic axes, the phase velocities of two or three plane waves coincide, and the behaviour of the degenerate waves becomes complicated in their vicinity. To calculate the acoustic axes in triclinic anisotropy, Vavryčuk (2005) solves the following Khatkevich equations (see Khatkevich 1963, eq. 11; Vavryčuk 2005, eqs 6a–c): with Γ_ik = a_ijkln_jn_l the Christoffel tensor, a_ijkl the density-normalized stiffness tensor and n_j the wave normal. However, only two of eqs (1a)–(c) are independent and solving them yields at most 36 isolated directions, among which only 16 directions are true acoustic axes, while the other 20 are spurious and should be eliminated. Considering that the Khatkevich equations are derived via multiplying the Darinskii equations by terms Γ₁₂Γ₁₃, Γ₁₂Γ₂₃ or Γ₁₃Γ₂₃, Vavryčuk (2005) claims that the solutions of equations (see his eqs 7a–c) or or are the spurious directions and should be eliminated from the complete set of solutions of the Khatkevich equations. Unfortunately, it is not accurate to call all the solutions of eqs (2a)–(c) spurious and simply eliminate them without further considerations. As shown below, not all the solutions of eqs (2a)–(c) must be necessarily spurious.

If two eigenvalues of the Christoffel tensor coincide (i.e. a P-S1 singularity or a S1-S2 singularity), the solution of eqs (2a)–(c) can be a true acoustic axis if one of the following conditions is satisfied: or or or

If only two of Γ₁₂, Γ₁₃ and Γ₂₃ vanish, for example, Γ₁₃ = Γ₂₃ = 0 and Γ₁₂ ≠ 0 (see eq. 4a), the three eigenvalues of the Christoffel tensor Γ_ik are given by If the last identity in eq. (4a) is satisfied, one obtains that Substituting eq. (6) into eq. (5), one obtains that Γ₃₃ is the degenerate eigenvalue and Γ₁₁ + Γ₂₂−Γ₃₃ is the non-degenerate one. As an example, we can mention anisotropy which possesses a longitudinal acoustic axis not coinciding with any of the coordinate axes but lying in a coordinate plane.

If three eigenvalues of the Christoffel tensor coincide in some direction (the so-called triple degeneracy), all three plane waves have identical phase velocities and the Christoffel tensor is isotropic: Obviously, eqs (2a)–(c) are satisfied in all coordinate systems in this case. An example of anisotropy with a triple degeneracy is the medium with elastic constants a₃₄ = a₃₅ = a₄₅ = 0 and a₃₃ = a₄₄ = a₅₅. The triple acoustic axis is along the x₃ axis. The axis is multiple and thus unstable.

The solutions of eqs (2a)–(c) satisfying the conditions of eqs (3), (4) and (7) should not be eliminated as proposed incorrectly by Vavryčuk (2005). Hence, the correct equations for determining the spurious directions are as follows: or or or